Initially, there are $3^{80}$ particles at the origin $(0,0)$. At each step, the particles are moved to points above the x-axis as follows: if there are $n$ particles at any point $(x,y)$, then $\lfloor n/3\rfloor$ of them are moved to $(x+1,y+1)$, $\lfloor n/3\rfloor$ are moved to $(x,y+1)$, and the remaining to $(x-1,y+1)$. For example, after the first step, there are $3^{79}$ particles each at $(1,1)$, $(0,1)$, and $(-1,1)$. After 80 steps, the number of particles at $(79,80)$ is:

Step 1: Let the distribution of particles at step $y$ be represented by a generating function. Initially, at step $0$, we have $3^{80}{z}^0$ at $y=0$.

Step 2: At each step, a group of $n$ particles at $(x,y)$ splits into three groups of sizes $\lfloor n/3\rfloor$, $\lfloor n/3\rfloor$, and the remainder. Since we start with $3^{80}$ particles and perform $80$ steps, at step $y$, all particle counts at occupied points are divisible by $3^{80-y}$. Therefore, at each step, the division is exact (there are no remainders because $n$ is always a multiple of 3), and each group gets exactly $n/3$ particles.

Step 3: The movement of a particle from $(x,y)$ to $(x+1,y+1)$, $(x,y+1)$, and $(x-1,y+1)$ corresponds to multiplying by the trinomial factor $\frac{1}{3}(z+1+z^{-1})$ for the x-coordinates.

Step 4: After $80$ steps, the generating function for the x-coordinates of the particles is:

Step 5: The number of particles at a point $(x,80)$ is the coefficient of $z^x$ in $G(z)$, which is the coefficient of $z^{x+80}$ in $(1+z+z^2{)}^{80}$.

Step 6: For the point $(79,80)$, we need the coefficient of $z^{79+80}=z^{159}$ in the expansion of $(1+z+z^2{)}^{80}$.

Step 7: By symmetry of the coefficients of $(1+z+z^2{)}^{80}$ (since the polynomial is palindromic of degree $160$), the coefficient of $z^{159}$ is equal to the coefficient of $z^1$.

Step 8: The expansion of $(1+z+z^2{)}^{80}$ begins with:

Thus, the coefficient of $z^1$ (and therefore of $z^{159}$) is exactly $80$.

Therefore, the number of particles at $(79,80)$ after 80 steps is $80$.